Download 3.4: Optimization Problems Activity Objective: Construct a box from a

3.4:
Optimization Problems
Activity
Objective: Construct a box from a given piece of paper.
Directions:
1. Take a card with a given dimension.
2. Cut squares with side length equal to that on your card from the four
corners of your paper.
3. Fold the 4 sides up to create an open topped box.
4. Find the volume of your box.
5. Do you think you could find a box with a larger volume? How?
What are the dimensions of the box if we cut 4 squares with side length x from
each corner?
What is the equation for the volume of the box?
What side length will give us a maximum volume?
volume?
What is the maximum
This is how we solve optimization problems.
Guidelines:
1. Identify all given quantities and all unknowns
2. Write a primary equation for the quantity to be maximized or minimized
3. Reduce the primary equation to one independent variable. This
sometimes requires the use of a secondary equation.
4. Determine the feasible domain.
5. Determine the desired maximum or minimum value by using the calculus
techniques we just learned.
Ex:
The product of two numbers is 72. Minimize the sum of the second number and
twice the first number.
Ex:
Find the points on the graph of y  4  x 2 that are closest to 0, 3.
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Ex:
A new homeowner has 600 meters of fencing to enclose a rectangular portion of
the backyard. What should be dimensions of the yard be to maximize the
enclosed area?
Ex:
A solid is formed by adjoining a hemisphere to one end of a right circular
cylinder. The total surface area of the solid is 1000 square centimeters. Find the
radius of the cylinder that produces the maximum volume.